Questions tagged [model-theory]

Model theory is the branch of mathematical logic which deals with the connection between a formal language and its interpretations, or models.

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Rigid non-archimedean real closed fields

Question. Is there a countable rigid non-Archimedean real closed field? Background: As usual, a structure is said to be rigid if the only automorphism of the structure is the identity map. It is ...
Ali Enayat's user avatar
33 votes
0 answers
2k views

Defining $\mathbb{Z}$ in $\mathbb{Q}$

It was proved by Poonen that $\mathbb{Z}$ is definable in the structure $(\mathbb{Q}, +, \cdot, 0, 1)$ using $\forall \exists$ formula. Koenigsmann has shown that $\mathbb{Z}$ is in fact definable by ...
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27 votes
0 answers
1k views

Where do uncountable models collapse to?

Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ...
Noah Schweber's user avatar
23 votes
0 answers
650 views

CH and automorphisms of ultrapowers of $\mathbb{Z}$ and $\mathbb{R}$

Notation and motivation. Given an algebraic structure $\mathbb{M}$ of cardinality at most the continuum and with countably many operations, and a nonprincipal ultrafilter $\cal{U}$ on a countably ...
Ali Enayat's user avatar
19 votes
0 answers
554 views

What algebraic properties are preserved by $\mathbb{N}\leadsto\beta\mathbb{N}$?

Given a binary operation $\star$ on $\mathbb{N}$, we can naturally extend $\star$ to a semicontinuous operation $\widehat{\star}$ on the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ as ...
Noah Schweber's user avatar
19 votes
0 answers
791 views

"Compactness for computability" - does it ever happen?

Throughout, "computable structure" means "first-order structure in a computable language with domain $\omega$ whose atomic diagram is computable." Say that a computable structure $...
Noah Schweber's user avatar
19 votes
0 answers
918 views

What is the Cantor-Bendixson rank of the space of first order theories?

Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its ...
Danielle Ulrich's user avatar
17 votes
0 answers
334 views

Ado's theorem and the reduction to positive characteristic

The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case? The ...
Dmitrii Korshunov's user avatar
17 votes
0 answers
1k views

Non-rigid ultrapowers in $\mathsf{ZFC}$?

Originally asked and bountied at MSE: Question: Can $\mathsf{ZFC}$ prove that for every countably infinite structure $\mathcal{A}$ in a countable language there is an ultrafilter $\mathcal{U}$ on $\...
Noah Schweber's user avatar
17 votes
1 answer
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How are the two natural ways to define “the category of models of a first-order theory $T$” related?

$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Elem{Elem}$Background/Motivation: Inspired by an interesting question by Joel, I’ve been wondering about the relationship between two very natural ...
John Goodrick's user avatar
14 votes
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Which functions have all the common $\forall\exists$-properties of continuous functions?

This is an attempt at partial progress towards this question. Meanwhile, Sam Sanders pointed out that my original term was already in use, as were a couple other back-up terms, so ... oh well. For a ...
Noah Schweber's user avatar
14 votes
0 answers
491 views

The Ax-Kochen isomorphism theorem and the continuum hypothesis

Let's recall that: (1): The Ax-Kochen principle says that if $\mathcal{U}$ is a non-principal ultrafilter over prime numbers, then $\prod_{\mathbb{U}} \mathbb{F}_p((t)) \equiv \prod_{\mathbb{U}} \...
Mohammad Golshani's user avatar
14 votes
0 answers
391 views

O-minimality and forcing

It is well-known that the structure $(\mathbb{R}, +, \cdot, <, 0, 1)$ is an o-minimal structure and hence the set of integers $\mathbb{Z}$ is not definable in it. In an ongoing project with Will ...
Mohammad Golshani's user avatar
14 votes
1 answer
582 views

On certain order-automorphisms of the rationals

Consider the rationals $\mathbb{Q}$ with the usual order $\leq$. Now let $A$ be a subset of $\mathbb{Q}$, such that foreseen with the induced order $\leq$, $(A,\leq)$ is a dense linear order. ...
THC's user avatar
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When does HSP reduce to SPH?

This is actually a poorly camouflaged attempt to use the answers to When is the opposite of the category of algebras of a Lawvere theory extensive? (all very interesting) for the purposes of my ...
მამუკა ჯიბლაძე's user avatar
13 votes
0 answers
679 views

Applications of Set theory vs. model theory in mathematics

I have a question that has occupied my mind for some time. Let's first consider applications of set theory and model theory in mathematics. Major applications of set theory are in topology, Banach ...
Mohammad Golshani's user avatar
13 votes
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226 views

Is there a finite equational basis for the join of the commutative and associative equations?

I asked this on math stack exchange, but I was told to post it on mathoverflow. Consider the lattice of equational theories of a single binary operation $*$. The meet of the theory axiomatized by the ...
user107952's user avatar
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Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories (In the 80s). Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following: (a) Trivial (...
Mostafa Mirabi's user avatar
12 votes
0 answers
473 views

Why is it so hard to give examples of differentially closed fields?

The theory of algebraically closed field, say in characteristic zero, and of differentially closed fields (of characteristic zero) have much in common: quantifier elimination and (hence) decidability; ...
Gro-Tsen's user avatar
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12 votes
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Sentences preserved under inverse limits

One of the classical theorems of model theory is the Chang-Łos-Suszko preservation theorem that states that the theories formulated in FOL (first order logic) that are preserved under direct limits (...
Ali Enayat's user avatar
12 votes
0 answers
240 views

Is there a characterization of the class of first-order formulas that are closed in every compact Hausdorff structure?

Fix a relational language $\mathcal{L}$. (I don't think relational really matters that much but I don't want to worry about it.) A topological $\mathcal{L}$-structure is an $\mathcal{L}$-structure $M$ ...
James Hanson's user avatar
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12 votes
0 answers
460 views

A question concerning model theory of groups

Several days ago, Professor Martin Bridson gave a very nice talk in my department. Some questions concerning his talk came into my brain Since I am neither a model theorist nor a algebraist, I am not ...
喻 良's user avatar
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12 votes
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Is there a Rado category?

The Rado graph appears to have a nice universality property (it contains all finite and all countably infinite graphs as induced subgraphs) and homogeinety property (any isomorphism between finite/...
Qfwfq's user avatar
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11 votes
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Example of $\aleph_1$-categorical linear order

Is it possible to have an $L_{\omega_1,\omega}$-sentence $\phi$ in a vocabulary that includes $<$ that satisfies the following? $<$ is a linear order on a definable subset; $\phi$ is $\aleph_1$-...
Ioannis Souldatos's user avatar
11 votes
0 answers
269 views

Maximality of linear orders in the Keisler order on theories

Recently Malliaris and Shelah (see their preprint http://math.uchicago.edu/~mem/Malliaris-Shelah-CST.pdf) have shown that theories with $SOP_2$ are maximal in the Keisler order. A preceding result of ...
matteo viale's user avatar
11 votes
0 answers
507 views

Using Lindstrom's theorem to prove Craig interpolation

[EDIT: The theorem I call "Beth definability" below is apparently not generally called that (wikipedia notwithstanding; see https://math.stackexchange.com/questions/288450/two-forms-of-beths-theorem). ...
Noah Schweber's user avatar
10 votes
0 answers
361 views

Can one define in ZFC a directed system of embeddings on the class of all linear orders realizing the surreal line as the direct limit?

Consider the surreal line $\langle\newcommand\No{\text{No}}\No,\leq\rangle$, in its order structure only. This is a proper class linear order, which is universal for all set-sized linear orders, as ...
Joel David Hamkins's user avatar
10 votes
0 answers
255 views

Classifying cohomology

In his 1973 topos seminar in Buffalo (the tapes are now freely available online!), Grothendieck said: The cohomology of a topos associated to an algebraic structure should be called the "...
LeopSchl's user avatar
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Extending models of topological set theory

$\mathsf{GPK_\infty^+}$ is an alternative set theory in which we have comprehension for formulas which are positive in a certain sense; see the SEP article for more detail (or this MO post, which ...
Noah Schweber's user avatar
10 votes
0 answers
316 views

Definability up to isomorphism versus definability of an isomorphic copy

Question: Is it provable in ZFC that every structure that is ordinal definable up to isomorphism has an ordinal definable isomorphic copy? If not, what are some counterexamples? All structures are ...
Dmytro Taranovsky's user avatar
10 votes
0 answers
398 views

Equational theory in the signature (+,*,0,1) of sedenions and beyond

Consider a Cayley-Dickson algebra $(X,+,∗,0,1)$, that is an algebra generated from the reals by the Cayley-Dickson construction. From complexes to quaternions, we lose commutativity of multiplication, ...
user107952's user avatar
  • 2,063
10 votes
0 answers
226 views

Is there an expansion of $(\mathbb{N},+,<)$ by a pairing function that is still NIP?

I was told once that there is a theory consisting of just a pairing function that is stable, although I cannot find a reference for it. This motivated my question, which is essentially the title, ...
James Hanson's user avatar
  • 10.3k
10 votes
0 answers
255 views

Collapsing the Linear Time Hierarchy and finite axiomatizability of bounded arithmetic

It is well known that if ${\bf T_2}$ (or $I\Delta_0+\Omega_1$) is finitely axiomatizable, then the Polynomial Hierarchy collapses. Q. Is there any similar relation between $I\Delta_0$ and Linear ...
Erfan Khaniki's user avatar
10 votes
0 answers
507 views

Riemann hypothesis in Zilber's field

Question. What is known about the situation (truth or falsity) of Riemann hypothesis in the Zilber's field?
Mohammad Golshani's user avatar
10 votes
0 answers
241 views

Two cardinal obstructions

Given a theory $T$ and a formula $\phi(x)$ we say that they admit a $(\kappa, \lambda)$ model if there is a model $M$ such that $|M| = \kappa$ and $|\phi(M)| = \lambda$. In all examples that I know ...
Levon Haykazyan's user avatar
10 votes
0 answers
349 views

Krull dimension and Morley rank

Definition : A Topological space $\mathcal{D}$ is called noetherian if it satisfies the descending chain condition for closed subsets. We define the dimension of $\mathcal{D}$ to be the supremum of ...
Mostafa Mirabi's user avatar
9 votes
0 answers
224 views

Continuum hypothesis analogue for substructures

This question was previously asked and bountied at MSE. Throughout, "theory" means "possibly-incomplete first-order theory in a countable language." Say that a theory $T$ has CHS (...
Noah Schweber's user avatar
9 votes
0 answers
265 views

Is “simplicity is elementary” still hard? (Felgner’s 1990 theorem on simple groups, and subsequent work)

I came across a reference in this MathOverflow answer to an intriguing result of Ulrich Felgner [1]: among finite non-Abelian groups, the property of being simple is first-order definable. According ...
Peter LeFanu Lumsdaine's user avatar
9 votes
0 answers
250 views

Can "$\exists\mathcal{X}(R\cong C(\mathcal{X}))$" be expressed in "large" infinitary second-order logic?

Originally asked and bountied at MSE without success: Say that a ring $R$ is spatial iff there is some topological space $\mathcal{X}$ such that $R\cong C(\mathcal{X})$, where $C(\mathcal{X})$ is the ...
Noah Schweber's user avatar
9 votes
0 answers
455 views

How many steps does it take to "Tarski-Vaughtify" second-order logic?

Given a regular logic $\mathcal{L}$, let $\preccurlyeq_\mathcal{L}$ be the usual elementary submodelhood relation for $\mathcal{L}$. There is also a separate submodelhood relation coming from the ...
Noah Schweber's user avatar
9 votes
0 answers
271 views

What logic characterizes relative intrinsic complexity in set recursion?

Short version: Is there an analogue of the Ash-Knight-Manasse-Slaman/Chisholm theorem for $E$-recursion? Long version: I'm interested in "$E$-recursive structure theory," but it's not ...
Noah Schweber's user avatar
9 votes
0 answers
220 views

Is there a ``Ladner's Theorem" for the PH-vs-PSPACE scenario?

Like a statement of the kind, ``If the Polynomial Hierarchy (PH) $\neq$ PSPACE then there exists $L \in PSPACE \backslash PH$ which is not PSPACE-complete"? Or is there something else that states ...
user6818's user avatar
  • 1,883
9 votes
0 answers
245 views

Is there a notion analogous to separability but requiring definable rather than countable sets?

Among models of $\lambda$-calculus, some like the Bohm tree model have the property that every element is a directed sup of definable elements, whereas others like the $D_\infty$ and $P(\omega)$ ...
fritzo's user avatar
  • 221
9 votes
0 answers
362 views

Is there Ultracoproduct-like construction for topological spaces in general?

In http://arxiv.org/pdf/math/9704205.pdf they define the ultracoproduct of a sequence of compact Hausdorff spaces, $\sum_\mathcal{U}X_i$ along an ultrafilter $\mathcal{U}$ as the Wallman-Frink ...
greg's user avatar
  • 191
9 votes
0 answers
278 views

Uncountable Lüroth problem

Question. Let $F(X)$ be the field obtained by adding an uncountable collection of indeterminates (mutually transcendental elements) to a prime field $F$. Is there an example of a subfield $E$ of $F(X)$...
Ali Enayat's user avatar
8 votes
0 answers
136 views

How strong is exponentiation with only open induction? (Or: "how low can we go?")

Do the strongest theories currently known to be unconstrained by Tennenbaum's theorem ($IOpen$ and some modest extensions) remain so when augmented with a definition of exponentiation and axiom $\...
Robin Saunders's user avatar
8 votes
0 answers
93 views

Is the hypotenuse operation associative in every Tarski plane?

By a Tarski space I understand a mathematical structure $(X,\mathsf B,\equiv)$ consisting of set $X$, a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\...
Taras Banakh's user avatar
  • 40.8k
8 votes
0 answers
238 views

First order formula describing connected components

I ask this question here after no answer came up in the original MathSE question. Let $\mathcal{L}$ be the language $\{+,-,\cdot,0,1,P\}$ where $P$ is some $n$-ary relation symbol. Is there a formula $...
Espace' etale's user avatar
8 votes
0 answers
164 views

Does determinacy imply unravellability for the Borel sets (over a weak base theory)?

As far as I know, the only way we currently know how to prove Borel determinacy in $\mathsf{ZFC}$ is to go through unravelability (a rather technical property whose definition can be found in Martin's ...
Noah Schweber's user avatar
8 votes
0 answers
177 views

Topological Vaught's conjecture for special theories

As is know, Vaught's conjecture is a special case of topological Vaught's conjecture. On the other hand, the Vaught's conjecture is true for the following theories: 1- $\omega$-stable theories (...
Mohammad Golshani's user avatar

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